Is there a way to "mirror" an empty?
Constraints may suit. The copy Xform constraints have the option to invert.

Here we have 2 axis-Empties, both parented to the cube-Empty in the middle. The right Empty is constrained to the left Empty as follows:



And it's parented to the cube Empty Without Inverse.

The arrows are parented to their respective axis-Empties, for illustration:



If the Empties are all ShiftD duplicated together, Blender is smart enough to keep the relations between the duplicates, and the whole rig can be moved/rotated using the cube.